The value of a such that the area bounded by the curve y = x 2 + 2 a x + 3 a 2 , the coordinate axes and the line x = 1 , attains its least value is equal to

Step-by-step solution

y = x 2 + 2 a x + 3 a 2 is positive for all x ∈ R Area = ∫ 0 1 x 2 + 2 a x + 3 a 2 d x = 1 3 + 2 a 1 2 + 3 a 2 = 1 3 + a + 3 a 2 Area = 9 a 2 + 3 a + 1 3 Which is minimum at a = - 1 6